PoS(CORFU2019)152 https://pos.sissa.it/ and I. Taskas ∗ Speaker. We discuss higher curvature (super) gravity intheory four which dimensions. contains a We describe massive the spin-twomassive spectrum ghost spin-two of and field the we at propose the a linearized way level. to turn it into a canonical ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Corfu Summer Institute 2019 "School and(CORFU2019) Workshops on Elementary Particle Physics and31 Gravity" August - 25 September 2019 Corfu, Greece I. Dalianis, A. Kehagias Physics Division, National Technical University of15780 Athens Zografou Campus, Athens, Greece Higher Curvature Supergravity PoS(CORFU2019)152 4 ε i (1) (2) = + ]. In 9 super- ] of the N 2 is a chi- 18 ] and it is , A. Kehagias 10 αβγ 17 W 1 supergravity , 16 = is the Ricci scalar R , N ) , -extended Weyl theory [ 0 2 N δ ε α , where = n ]. This models propagates the + ( R Weyl 11 αβγ with five degrees of freedom and  + W + ˙ R δ µν R 2 g D δβαρ m , spinor indices. The spin of the highest W ˙ γ γ ρ ) ˙ β ε ˙ C α ε supergravity has been studied [ β , ε 2 2 theory. The super-Weyl tensor W ( 1 4 R 0 of the 2 = SL − + 1 8 of Poincare supergravity corresponds to ∗ → theory, which contains also the standard Einstein- power of the Weyl, Riemann and Ricci tensors and ˙ ) ε α R = 2 m A th ˙ ε n αβγ ε theory, inflation is driven entirely by the gravity itself, N 2 D W ]. This class of models has also been explored in [ is a fermionic spin-3/2 field and therefore, the spin of the ( 1 supergravity, and its spectrum in has been determined in R 8 γδ , ε , + = ) i 7 4 αβγ R denotes , ]. These theories are very interesting although they propagate  W 6 αβγ n N ], the inclusion of quadratic curvature terms in the action makes 2 are standard ( , 1 6 21 , ]. 1 denotes the additional five terms from the symmetrization of the R 5 1 , W , ) 31 = [ 4 = 20 | = ]. It should be noted here that such states leads to the impossibility of , 1 , γ α 3 theory should also include only the square of the Weyl tensor which is , . It also contains the spin-two field δεα δ ε α , 19 αβγ ) 2 β ( 2 , 0 W are the Weyl tensor and the vector auxiliary of the R and W , , γ ]. α 1 3 2 ˙ ε α ε + D ( 15 , A R δ and ]. 14 δβαρ W Here we will describe the massive Weyl Let us now discuss the linearized super-Weyl An interesting limit is the massless limit However, the Higher curvature theories are theories of the general form ] and [ 13 Weyl supergravity [ (in spinorial notation). fermionic indices Hilbert action, and itsgravities. supersymmetric The extensions. inclusion of The the Einstein-Hilbert lattermassless action are can theory. be In the regarded addition, as the a maximal mass deformation of the choice, opposite to the stanfdard one,are one propagaiting may forward in achieve unitarity time. but in this case, negative energies ral superfield, where component of the Weyl superfield it satisfies Its bosonic content follows from where without the need for antheory extra can scalar easily be and embedded thus in making the theory quite appealing. In addition, the ghost sates as in thesuperconformal theories massless with limits, four derivative i.e. terms. no Einstein-Hilbert term, they are simple examples of Weyl multiplet is is maintain at one shot both unitarity and propagation of positive energy states. Indeed, using a particular, as has been noted inthe [ theory renormalizable.ghost But states. there is This a ghosttensor. price state A to originates particular be from class paid the of namely such square models, the of the appearance the Riemann, ofusual massless the a graviton Ricci and massive a or massive spin-0 thethe state, inflaton Weyl the field. “scalaron". The Therefore, latter in can the be identified with [ also quadratic in the curvatureto since the terms square quadratic of in theinclusion the Weyl Riemann of or (or the the Ricci) scalar Weyl tensor curvatureghost tensor are spin-2 by now related massive the leas state 4D [ to Gauss-Bonnet a topological considerable term. problem: The the theory now propagates a conformal supergravity [ Highe Curvature SUGRA connected to the inflationary dynamics of the Starobinsky model [ describing Einstein gravity and the curvature scalar [ PoS(CORFU2019)152 ,  1 − 4) µν (8) (6) (3) (4) (7) (5) , ν F = 3 A , µν µ 2 , ∂ N F charges, 1 4 3 R = ) A. Kehagias = − 1 ( µν F N U 20 dof. µνρσ R q = W F n and 1 Weyl supergravity. It µνρσ + h B = ., W n c .  . c ) 2 N supergravities ( 1 g 1 representations. + ( 2 2 2 ) 2 + ( W , Spin  ) τ with helicity 2 1 E USp ) + . ) 2 + 2 ) R µνρσ , / 0 Θ q ε 3 2 , , 2 3 . ( 2 h d is the auxiliary scalar. Note however, that the ρσ ( α i − F u Z -extended Weyl + 4 µν Spin ]: 2 ) + (+ F 2 N and ) + ( + 0 1 × g 2 3 17 , 1 2 µ 2 , 2 − A = − ER 16 dof: 16 , (+ τ ) + 2 2 3 2 HP 2 : = ( R Θ − 1 gravity theory contains a total of 2 F 1 2 ( = n d 2  N is the topological Hirzebruch-Pontryagin term, + Z 2 g α B : Spin = n 1 µν + = ρσ u W the complex coupling , 1 supergravity are arranged in u N W Φ τ 1 3 w = ] with L ) turns out to be − 3 κλρσ µ 33 is dynamical in Weyl supergravity. N ε A µ µ A A 3 1 µνκλ 4 dof: W + = R = F 1 2 n 2 HP = + R B is the field strength of the "auxiliary" W n Similarly, the spectrum of the all existing It turns out that the supersymmetric action of the massive Weyl gravity is µ L 1 A ν − Therefore, the massive super-(Weyl) The massive states in with together with its CPT conjugate in 4D can besummarized determined. in the The following physicalconjugate table. are spectrum denoted in of The the the first top column various helicities [ Weyl-supergravity of theories each is multiplet together with its CPT e Highe Curvature SUGRA where we have denoted as The bosonic sector of ( ∂ vector "auxiliary" It is now straightforward tocontains determine the the following spectrum states: of the massive (i) The standard massless spin-2 graviton multiplet (ii) A massive spin-twosuper-Weyl supermultiplet multiplet [ in the non-standard sector. This is the massive where PoS(CORFU2019)152 (9) (12) (14) (11) (13) (10) -term 2 A. Kehagias 1 = 1 1 1 2 2 1 , N  R 2 2 . κ is the Weyl tensor. The Weyl = Ω 2 2 2 2 1 1 + ν ν ] σ N ∇ , , g µ 0 ρ µν . [ µνρσ ∇ g 3 µ = 2 g W µν = νρσ 3 µ 2 2 3 3 1 1 Ω g µν / 3 W G R = − N 3 µνρσ 2 Ω = − a κ µν W 6 ν 2 4 b g a 2 ] − σ  νρσ = µ + 2 2 4 4 R → R g b 2 ρ W [ µν − N − µ µν B √ g g Ω x 4 − 2 2 2 2 = d 2 2 2 2 2 2 1 / / / 1 / ν ] b 0 / 0 / 0 1 2 / 1 / 2 / 1 2 / 2 R 1 3 1 1 gravity theory in four dimensions is described by has the following − − ρ 1 1 3 3 3 3 Z 2 − − − − ======R ======σ = = = = [ µ S max max max max max max max CPT max CPT max CPT max CPT max CPT max g h h max h max h max h h max h h CPT max h CPT max h h h CPT max CPT max CPT max CPT max h h h h h h h h h h − µνρσ R = µνρσ W The bosonic massive Weyl The eom can be written as action Contrary, the Einstein-Hilbert term transforms under conformal transformations, as is conformal invariant since under the conformal transformation the Weyl tensor is inert and thus, it can be regarded as a mass term in the theory. Highe Curvature SUGRA where PoS(CORFU2019)152 ) µν 19 w (19) (15) (18) (20) (21) (22) (16) (17) A. Kehagias .  ) 2 term and it is is a ghost ) since aw 2 . 18 ,  − and the last term describes 2  R µν µν , 2 R ) w c g 2 µν . . Consider for example the action µν + R g 1 3 w 6 1 . µν ( term is absent and in this case, ( w 2 − W − ρµσν − 1 µνρσ g ) . 2 W . It can be shown that the two-derivative 3 is given by µν g 0 R W µν ( and 1 R ρσ µν 1 . w = µν ) µν R − w µν B µν − g µν R 2 1 a . It is due to the Weyl µνρσ R ( g µν a B ( 4 a  ). Indeed, using the equation of motion for B 2 W + 2 / µν 1 µ − 2 . W = 4 18 c 1 G + ∇ g κ Ω 2 µρν µν σ + c W 2 g = = g , R GB W 0 2 P , σ µν µν = M 2 = W ) + ∇ b B w 1 g g  ρ µ ( 2 µ g ∇ R B with mass − µνρσ 2 P = = ⇒ √ 1 W M µν x c  µν 4 w gravity theory contains seven propagating dof. g is the Einstein tensor for the metric d 0. This is the non-standard sector of the theory. B 2 − µνρσ µν S < M µν W √ w δ Z a x δ 4 Rg = d 2 S / 1 is classically equivalent to ( M 1 the scalar mode associated to the Z ] − 1 µν = = ), it turns out that the theory is classically equivalent to ( w µν 2 massless standard graviton a S R 19 ± = is the Einstein tensor and the Bach tensor is the Gauss-Bonnet term. In fact we find that µν µν 0 or a tachyon for G GB G > An alternative way to identify the spectrum of the bosonic massive Weyl theory is to express To find the spectrum of the theory one should determine the poles in the propagators. It turns a a mass term for describes massive Weyl gravity. action where The Bach tensor transforms covarinatly under conformal transformations (ii) A massive spin-two particle Now, let us introduce a second symmetric tensor field in the action ( where Therefore, for (i) A helicity- Therefore, the massive (Weyl) it as a bimetric gravity theory with two spin-2 fields Highe Curvature SUGRA where It is also symmetric anddiff-invariance traceless due to conformal invariance as well as divergence-free due to for out that we have [ PoS(CORFU2019)152 ab w i b ∇ (27) (26) (29) (31) (24) (28) (25) (23) a ∇ 0 (30) 1). Note . In order µν . = g ) a term of = A. Kehagias µν ,  + 19 i  w α  2 )  . ab   g w 2 ( ab w ab b w (for indicating that the − o w g are now covariant µν ∇ ) may be turned into ab − a µν µν G αβ µν µν µν E g h E ∇ w 19 w w µν β − µν − ∂ w µν α w ab w µν ∂ cw g   w ν − + around Minkowski spacetime   − ∇
h µ 2 − µν µν  αβ w ∇ g  η w αβ − + − h − µν b αβ µ µν w , η µν δ ν αβ E µν µν a w is a second massive spin-2 tensor field. µν − g ∇ E ∇ µν w h µ µν ν ν = 2 µν w P µν 2 P ∇ ab w ∂ ∇ 2 P w w M ) of the kinetic term of µ w M 2 2 + µν − M ∂ W 2 w W 1 1 h a − ab µν b µν ) g g ν g 2 W 5 + h ν E δ g µν ) − a is implicit in the in mixed term + − w α ν together with massive spin-2 a µν w + ∇ h < µν µν µ ∇ αβ α µν c cw h h µν µ ) turns out to be written as αβ w ∂ ∇ ( G g h µ ( = 23 ∇ αβ − w µν 2 ∂ 2 g b αβ ν E 2 µν µν δ − E + h a µν µ − in the previous action. The new EOM read h δ R µν µν 2 P µν 2 P h  w 2 µν h 2 is as usual the tensor P M  g M  2  c 2 M h −  αβ n µν c 2 g −  + E 2 1 g −  − ab − g − g √ x and µν − = √ 4 µν x P d √ g 4 wg x αβ M d 4 − Z h / + d b ν Z is clearly the massless graviton and W = αβ δ µν µν Z a M µ = S E w ). Then we find that to quadratic order we have µν ) describes a massless spin-2 δ 2 = = h h − µν 2 19 S W η = S µν g ] to be equivalent to adding an infinite number of higher derivative terms. G µν Another way to remove the ghost stae is the following. Let us add to the action ( Eq.( w 34 G 1 g w 1 g with The field second spin-2 tensor mode is actuallya a ghost-free ghost. theory The by bimetric the gravity inclusion theoryin of ( [ an infinite number of terms. This procedure has been shown the form Notice that the derivativesderivatives and and the the full metric metric that appear in the definition of or Note however that the wrong sign (compared to to explicitly see this, let us consider fluctuations Let us now define a new field This term clearly removes thespin-2 particle ghost, with and, canonical kinetic thus term. the The theory action will in this contain case a reads graviton and a massive Highe Curvature SUGRA that the kinetic term for the massive spin-2 (metric in tersm of which, the quadratic action ( where PoS(CORFU2019)152 ab w i b ∂ (33) (38) (41) (32) (36) (37) (39) (40) (34) (35) a # ∂  ab µν w w η  ab A. Kehagias  κλ + h E − ) becomes i , ). Linearizing   i ab κλ µν 35 32 ab
w ⊥ w µν 2 η b η w w µν ∇ w ¯ h µν a g +  µν − w ⊥ η ) µν c
2 ) and ( ∇ µν w  ν i η B κλ µν ) − η + + ∇ 31 h − 1 4 w 4 1  D µν 0 D µ λ ab  ( ab ⊥  2 + ∂ − µν  η + SS ∇ w w κ ν w ¯ h w − D B 3 4 +  ∂ ∂ + µν ⊥ − µ  w   w − (  µν ∂ − w µν ab ab   στ µ
η =  + ab − w E ⊥ B 0 ) µν µν ) a − Rw µ w S ab C  λτ η η D δ h D ab w g −  b 1 4 µν 2 1 4 P ν −    , ⊥ ∂ µ E ∂ w ¯ h M κσ 2 − ν − µ − w C + g 2 w 2 ∂ µν ∂ ν µν 2 ν g g w  3 8 w ∂ ∂  D w R ( − − + µ µ − is transverse. Then the action ( 2 φ P +  b )( 3 8 − ν ∂ µν + ∂  c δ ⊥  0 remains a solution to ( δ ab ν M µν g D 1 µ  − σ  S 6 a − h µν 2 w ⊥ φ C w δ  = ¯ = w ¯ ∂ g h w  + 8 3 +   ν ab µ µν ⊥ S 2 P −  ⊥ µ µ ⊥ ∂ − = µν ∇ G − 3 8 − w κλ M w A C 1 σ  w µν S − ( ab  2 w ν ν − µν ⊥ µλ ¯ ⊥ ¯ ∇ h b cw g ∂ ∂ ν 3 4 ¯ ¯ µν h h h µν 2 δ , ab − g µν ⊥ − λ µν + + a ⊥   µ − − E σ ν w ∂ δ ⊥ w ⊥ ν ν 2 + P µ ν µν  w 4 B  A
C  ⊥  µν ∂ M ⊥ µ a µ µ ¯ as h   ¯ 2 µν C h P h µ µν ∂ ∂ + ∇ 2 P x along with c 2 ⊥   w − M 2 σ 4 µν + + 1 M a ν ⊥ ww µ 2 w w ¯ νλ 2 h d P ∇ + µν w g 4 h G C  − µν µν ⊥ ⊥ M − η 2 ¯ λ = 2 Z − ab ¯ ) 2 h P h h + w ∂ a ν η = x φ x and a µ µν + + = M 4 4 = w ν = 1 ∂ 2 w µν d w d ) µ µν w µν ( g 2  η g ( µν a  are transverse traceless and ¯ µ a µν h 2 ¯ Z Z ¯ 1 2 S h " w − w G b  w µν = ⊥ ν = + = 1 g ) 2 δ 2 w ) 2 a µ 2 ( µν ( δ S + − = S h G and ) turns out to be δ = µν 39 µν ⊥ G ¯ h µν 2 P G Flat space-time M ) around this background yields δ w 1 31 g with where We can decompose We introduce the following scalars so that ( Highe Curvature SUGRA and ( The linearized action is then given by where PoS(CORFU2019)152 i µν ⊥ w (42) (43) µν ⊥ and a w ⊥ µ − C µ A. Kehagias ⊥ (1985) 1169. (1977) 953. C 31  16 ⊥ µ C 2 +  (1978) 353. 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